Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004

Outline Definition The encoding process Decoding population codes Quantifying information: Shannon and Fisher information Basis functions and optimal computation

Outline Definition The encoding process Decoding population codes Quantifying information: Shannon and Fisher information Basis functions and optimal computation

Receptive field s: Direction of motion Stimulus Response Code: number of spikes 10

7 8 4 Receptive field s: Direction of motion Trial 1 Stimulus Trial 2 Trial 3 Trial 4

Variance of the noise,  i (  ) 2 Encoded variable (s) Mean activity f i (  ) Variance,  i (s) 2, can depend on the input Tuning curve f i (s)

Tuning curves and noise Example of tuning curves: Retinal location, orientation, depth, color, eye movements, arm movements, numbers… etc.

Population Codes Tuning CurvesPattern of activity (r) Direction (deg) Activity Preferred Direction (deg) Activity s?s?

Bayesian approach We want to recover P(s|r). Using Bayes theorem, we have:

Bayesian approach Bayes rule:

Bayesian approach We want to recover P(s|r). Using Bayes theorem, we have: likelihood of s posterior distribution over s prior distribution over r prior distribution over s

Bayesian approach If we are to do any type of computation with population codes, we need a probabilistic model of how the activity are generated, p(r|s), i.e., we need to model the encoding process.

Activity distribution P(r i |s=-60) P(r i |s=0) P(r i |s=-60)

Tuning curves and noise The activity (# of spikes per second) of a neuron can be written as: where f i (  ) is the mean activity of the neuron (the tuning curve) and n i is a noise with zero mean. If the noise is gaussian, then:

Probability distributions and activity The noise is a random variable which can be characterized by a conditional probability distribution, P(n i |s). The distributions of the activity P(r i |s). and the noise differ only by their means (E[n i ]=0, E[r i ]=f i (s)).

Gaussian noise with fixed variance Gaussian noise with variance equal to the mean Examples of activity distributions

Poisson distribution: The variance of a Poisson distribution is equal to its mean.

Comparison of Poisson vs Gaussian noise with variance equal to the mean Activity (spike/sec) Probability

Gaussian noise with fixed variance Population of neurons

Gaussian noise with arbitrary covariance matrix  : Population of neurons

Outline Definition The encoding process Decoding population codes Quantifying information: Shannon and Fisher information Basis functions and optimal computation

Population Codes Tuning CurvesPattern of activity (r) Direction (deg) Activity Preferred Direction (deg) Activity s?s?

Nature of the problem In response to a stimulus with unknown value s, you observe a pattern of activity r. What can you say about s given r? Bayesian approach: recover p(s|r) (the posterior distribution) Estimation theory: come up with a single value estimate  from r

Estimation Theory Preferred orientation Activity vector: r Decoder Encoder (nervous system)

Preferred retinal location r 200 Decoder Trial 200 Encoder (nervous system) Preferred retinal location r2r2 Decoder Trial 2 Encoder (nervous system) Preferred retinal location r1r1 Decoder Trial 1 Encoder (nervous system)...

Preferred retinal location r DecoderEncoder Estimation Theory If, the estimate is said to be unbiased If is as small as possible, the estimate is said to be efficient

Estimation theory A common measure of decoding performance is the mean square error between the estimate and the true value This error can be decomposed as:

Efficient Estimators The smallest achievable variance for an unbiased estimator is known as the Cramer- Rao bound,  CR 2. An efficient estimator is such that In general :

Estimation Theory Preferred orientation Activity vector: r Decoder Encoder (nervous system) Examples of decoders

Voting Methods Optimal Linear Estimator

Linear Estimators

X and Y must be zero mean Trust cells that have small variances and large covariances

Voting Methods Optimal Linear Estimator

Voting Methods Optimal Linear Estimator Center of Mass Linear in r i /  j r j Weights set to s i

Center of Mass/Population Vector The center of mass is optimal (unbiased and efficient) iff: The tuning curves are gaussian with a zero baseline, uniformly distributed and the noise follows a Poisson distribution In general, the center of mass has a large bias and a large variance

Voting Methods Optimal Linear Estimator Center of Mass Population Vector

s riPiriPi P

Voting Methods Optimal Linear Estimator Center of Mass Population Vector Linear in r i Weights set to P i Nonlinear step

Population Vector Typically, Population vector is not the optimal linear estimator.

Population Vector

Population vector is optimal iff: The tuning curves are cosine, uniformly distributed and the noise follows a normal distribution with fixed variance In most cases, the population vector is biased and has a large variance

Maximum Likelihood The maximum likelihood estimate is the value of s maximizing the likelihood P(r|s). Therefore, we seek such that: is unbiased and efficient. Noise distribution

Maximum Likelihood Tuning Curves Direction (deg) Activity Pattern of activity (r) Preferred Direction (deg) Activity

Maximum Likelihood Template

Preferred Direction (deg) Activity Maximum Likelihood Template

ML and template matching Maximum likelihood is a template matching procedure BUT the metric used is not always the Euclidean distance, it depends on the noise distribution.

Maximum Likelihood The maximum likelihood estimate is the value of s maximizing the likelihood P(r|s). Therefore, we seek such that:

Maximum Likelihood If the noise is gaussian and independent Therefore and the estimate is given by: Distance measure: Template matching

Maximum Likelihood Preferred Direction (deg) Activity

Gaussian noise with variance proportional to the mean If the noise is gaussian with variance proportional to the mean, the distance being minimized changes to: Data point with small variance are weighted more heavily

Bayesian approach We want to recover P(s|r). Using Bayes theorem, we have:

Bayesian approach The prior P(s) correspond to any knowledge we may have about s before we get to see any activity. Note: the Bayesian approach does not reduce to the use of a prior…

Bayesian approach Once we have P(s  r), we can proceed in two different ways. We can keep this distribution for Bayesian inferences (as we would do in a Bayesian network) or we can make a decision about s. For instance, we can estimate s as being the value that maximizes P(s|r), This is known as the maximum a posteriori estimate (MAP). For flat prior, ML and MAP are equivalent.

Bayesian approach Limitations: the Bayesian approach and ML require a lot of data (estimating P(r|s) requires at least n+(n-1)(n-1)/2 parameters)…

Bayesian approach Limitations: the Bayesian approach and ML require a lot of data (estimating P(r|s) requires at least O(n 2 ) parameters, n=100, n 2 =10000)… Alternative: estimate P(s|r) directly using a nonlinear estimate (if s is a scalar and P(s|r) is gaussian, we only need to estimate two parameters!).

Outline Definition The encoding process Decoding population codes Quantifying information: Shannon and Fisher information Basis functions and optimal computation

Fisher information is defined as: and it is equal to: where P(r|s) is the distribution of the neuronal noise. Fisher Information

For one neuron with Poisson noise For n independent neurons : The more neurons, the better!Small variance is good! Large slope is good!

Fisher Information and Tuning Curves Fisher information is maximum where the slope is maximum This is consistent with adaptation experiments Fisher information adds up for independent neurons (unlike Shannon information!)

Fisher Information In 1D, Fisher information decreases as the width of the tuning curves increases In 2D, Fisher information does not depend on the width of the tuning curve In 3D and above, Fisher information increases as the width of the tuning curves increases WARNING: this is true for independent gaussian noise.

Ideal observer The discrimination threshold of an ideal observer,  s, is proportional to the variance of the Cramer-Rao Bound. In other words, an efficient estimator is an ideal observer.

An ideal observer is an observer that can recover all the Fisher information in the activity (easy link between Fisher information and behavioral performance) If all distributions are gaussian, Fisher information is the same as Shannon information.

Population Vector and Fisher Information Population vector CR bound Population vector should NEVER be used to estimate information content!!!! The indirect method is prone to severe problems… 1/Fisher information

Outline Definition The encoding process Decoding population codes Quantifying information: Shannon and Fisher information Basis functions and optimal computation

So far we have only talked about decoding from the point of view of an experimentalists. How is that relevant to neural computation? Neurons do not decode, they compute! What kind of computation can we perform with population codes?

Computing functions If we denote the sensory input as a vector S and the motor command as M, a sensorimotor transformation is a mapping from S to M: M=f(S) Where f is typically a nonlinear function

Example 2 Joint arm:  x  y

Basis functions Most nonlinear functions can be approximated by linear combinations of basis functions: Ex: Fourier Transform Ex: Radial Basis Functions

Basis Functions Direction (deg) Activity Preferred Direction (deg) Activity

Basis Functions A basis functions decomposition is like a three layer network. The intermediate units are the basis functions X y

Basis Functions Networks with sigmoidal units are also basis function networks

Basis Function Layer AB CD XY Z 23 Y Z Z Z X YX XY Linear Combination YX YX YX YX

Basis Functions Decompose the computation of M=f(S,P) in two stages: 1.Compute basis functions of S and P 2.Combine the basis functions linearly to obtain the motor command

Basis Functions Note that M can be a population code, e.g. the components of that vector could correspond to units with bell-shaped tuning curves.

Eye Position: X e Head position Gaze + Fixation point Head-centered Location: X a Retinal Location: X r Example: Computing the head-centered location of an object from its retinal location

Basis Functions

H k =R i +E j Preferred retinal location Preferred eye location Preferred head centered location RiRi EjEj Basis Function Units Gain Field Activity Eye-centered location E=20° E=0° E=-20°

H k =R i +E j Preferred retinal location Preferred eye location Preferred head centered location RiRi EjEj Basis Function Units Partially shifting receptive field Activity Eye-centered location E=20° E=0° E=-20°

Fixation point Head-centered location Retinotopic location Screen Visual receptive fields in VIP are partially shifting with the eye (Duhamel, Bremmer, BenHamed and Graf, 1997)

Summary Definition Population codes involve the concerted activity of large populations of neurons The encoding process The activity of the neurons can be formalized as being the sum of a tuning curve plus noise

Summary Decoding population codes Optimal decoding can be performed with Maximum Likelihood estimation (x ML ) or Bayesian inferences (p(s|r)) Quantifying information: Fisher information Fisher information provides an upper bound on the amount of information available in a population code

Summary Basis functions and optimal computation Population codes can be used to perform arbitrary nonlinear transformations because they provide basis sets.

Where do we go from here? Computation and Bayesian inferences Knill, Koerding, Todorov: Experimental evidence for Bayesian inferences in humans. Shadlen: Neural basis of Bayesian inferences Latham, Olshausen: Bayesian inferences in recurrent neural nets

Where do we go from here? Other encoding hypothesis: probabilistic interpretations Zemel, Rao